180 



Obviously, frictional resistance must control the currents when a 

 connecting or a closed canal approaches its critical lengths. 



348. Nodes in a closed canal. — The equation of a component of the 

 tide at a point in a closed canal distant x from the entrance is, equation 



(251): 



y=Ao cos {at-\-ao) cos (l—x/L)y/cos y 



The tide, y, becomes zero, for all values of t, at the points at wliicb 

 {l—x/L)y = Tr/2, 37r/2, 5ir/2, etc. At these points x=L—TrL/2y, 

 L-3TrL/2y, L-5TL/2y, etc., and hence x=L-\/4, i-3X/4, i-5X/4, 

 etc. 



If the flow were frictionless and the canal long enough, each com- 

 ponent of the tide would then disappear at points one-quarter, three- 

 quarters, etc., of its wave length from the head of the canal. At 

 these points the component current would reach a maximum ampli- 

 tude of (^/c)^o/cos7. These points are termed nodal points. 



Similarly a component current in a closed canal would become zero 

 at the points at which sin (l—x/L)y=0, and hence at which: 



x=L, 1 — /2X, L—\, etc. 



And at these points the component of the tide would have a maxi- 

 mum amplitude of Aojcos y. 



It is perhaps needless to point out that true nodal points have no 

 counterpart in actual channels. 



349. Shallow water components of Jrictionless tides and currents. — 

 The variation of the mean depth, D, in the equation of continuity 

 (equation 184) with the rise and fall of the tide has been neglected in 

 the preceding analysis of the tides and currents in a canal without 

 frictional resistance. This variation must in fact produce distortions 

 of the simple harmonic fluctuations of the components of the tides 

 and currents that have been deduced. It may be shown that these 

 distortions, like those due to the form of the friction term, may be 

 reproduced by overcurrents and compound currents, and overtides 

 and compound tides. As iUustrated in the cubature of the Delaware 

 River, in chapter VI, this variation in the depth may produce marked 

 distortions of the current in a long and comparatively shallow channel; 

 but a mathematical analysis of the distortion with frictionless flow 

 does not serve much useful purpose. 



SEICHES 



350. An accidental tilting of the surface of a deep lake or enclosed 

 sea, such as may be produced by wind, or a variation in the barometric 

 pressure, or by any other cause, often is followed by periodic oscilla- 



